Let be a group with normal subgroups and .
Let .
Suppose $M \cap \langle a \rangle =1$ and $N \cap \langle a \rangle= \langle a^{s} \rangle$ for some positive integer .
Can we find $MN \cap \langle a \rangle$???
I am not entirely sure what you are asking...I mean, given finite time of course you can! On the other hand, there isn't a definitive answer. For example, take,
( denoting the cyclic group of order ), with your elements being of the form with . Now, let . Clearly, if then , while if then . Clearly, . On the other hand, if we let then contains but still...so, basically, it depends on and and , and can be very different depending on different choices...